Constructing continuous-variable spacetime quantum states from measurement correlations. (arXiv:1903.06312v1 [quant-ph])

Quantum states, no matter pure states, density matrices or Wigner functions,
fully describe a physical system at a particular time. Time remains to be one
of the deepest mysteries in physics for centuries. Here we attempt to
understand time via a unified approach on both space and time, under the
intuition that relativity treats space and time on an equal footing; thus we
build quantum states across spacetime instead of only on spatial slices. We no
longer distinguish measurements on the same system at different times with
measurements on different systems at one time and construct spacetime states
upon these measurement statistics. As a first step towards quantum field
theory, we consider how to approach this in the continuous-variable multi-mode
regime. We propose six possible definitions for spacetime states in continuous
variables, based on four different measurement processes: quadratures,
displaced parity operators, position measurements and weak measurements. They
are motivated by the pseudo-density matrix formulation among indefinite causal
structures and the path integral formalism. We show that these definitions lead
to desirable properties, and suggest the differences and similarities between
spatial and temporal correlations. An experimental proposal for tomography is
suggested to demonstrate that the spacetime states we define have very strong
operational meanings. All these show that it is possible to define a
continuous-variable spacetime state with multiple desirable properties and with
the possibility to apply the arsenal of continuous-variable quantum state tools
to the analysis thereof.

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